高等代数中有关问题的降价算法

1 行列式值的计算求行列式值常用的计算方法[1]多种多样,但对高阶行列式计算很少有通用的公式．在计算机普及的条件下,公式法显得尤为重要,下面给出一种简单易行的公式法--降阶算法,它充分利用了二阶行列式计算简单、可以心算的特点,逐次降阶而又不增子行列式计算的个数,减少了计算量又易于上机,从而提高了计算速度．     

四阶行列式对角线法则的图解法

利用图解法,根据二阶和三阶行列式的对角线法则,通过总结规律,推导出四阶行列式的对角线法则,提供一种四阶行列式展开的图解方法,使普通四阶行列式的计算更具有可操作性.     

四阶行列式对角线法则的图解法

利用图解法,根据二阶和三阶行列式的对角线法则,通过总结规律,推导出四阶行列式的对角线法则,提供一种四阶行列式展开的图解方法,使普通四阶行列式的计算更具有可操作性.     

介绍一种用四分块矩阵计算n阶行列式的方法

介绍一种用四分块矩阵计算n阶行列式的方法,这种计算方法对某些n阶行列式是较为有用的一种方法,它适用于比较复杂特殊的行列式.     

计算n阶行列式的一种方法

利用分块矩阵的行列式的运算规则,证明一个条件等式及其推论,并给出计算n阶行列式的一种方法．     

范德蒙行列式的新证明及其应用

首先给出范德蒙行列式的一个新的证明,与以前的证明相比,本文给出的证明更具有通用性．其次介绍范德蒙行列式在n阶行列式计算和在向量组线性相关性证明中的两个应用．     

介绍一种用公式计算行列式的方法

在高等代数课程中,对n阶行列式的计算,一般方法是不存在的,但处理特殊类型的行列式有各种不同的方法。本文介绍一种用公式计算行列式的方法,这种计算方法对某种n阶行列式是较为有用的一种方法,它适用于较复杂     

介绍两种行列式的计算方法

<正> 在高等代数课程中,n阶行列式的计算是一个主要内容。但是n阶行列式的计算还没有一个普遍适用的方法、在处理特殊类型的行列式有各种不同的方法。本文将给出:两种行列式的计算方法,使用这两种方法不仅可以解决用其他方法难以计算出的行列式,并且极大简化     

Vandermonde行列式的快速并行算法

n阶Vandermonder行列式的求值通常需要O(n~2)次算术运算．本文从计算复杂性的角度出发，给出一种求Vandermonde行列式、合流型Vandermonde行列式、广义Vandermonde行列式的快速算法．该算法仅需O(nlog~2n)次算术运算．若在n台处理机上并行计算，该算法需并行步数O(nlog_(2~2)n)．速度倍数为s_p=O(n)．并行效率为O(1)．     

Wronsky行列式及其一阶微分的关系

证明了n阶行列式一阶微分为其逐次微分之和,进而得到了Wronsky行列式及其一阶微分的关系．     

Equations aux différences finies et déterminants d'intégrales de fonctions multiformes

We give a formula which computes the determinant of a matrix whose entries are integrals of algebraic differential forms multiplied by a product of complex powers of polynomials. In this formula enters the characteristic polynomial of some monodromies associated with this family of polynomials.     

on hybrid preconditioning methods for large sparse saddle-point problems

Based on the block-triangular product approximation to a 2-by-2 block matrix, a class of hybrid preconditioning methods is designed for accelerating the MINRES method for solving saddle-point problems. The appropriate values for the parameters involved in the new preconditioners are estimated, so that the numerical conditioning and the spectral property of the saddle-point matrix of the linear system can be substantially improved. Several practical hybrid preconditioners and the corresponding preconditioning iterative methods are constructed and studied, too.     

Sums of nilpotent matrices

We study matrices over general rings which are sums of nilpotent matrices. We show that over commutative rings all matrices with nilpotent trace are sums of three nilpotent matrices. We characterize 2-by-2 matrices with integer entries which are sums of two nilpotents via the solvability of a quadratic Diophantine equation. Some exemples in the case of matrices over noncommutative rings are given.     

构造正交向量的一个新方法及其应用

利用向量的数量积及行列式的按行(列)展开定理,构造出一个n维向量,它能够与n-1个n维向量都正交.这种构造正交向量的方法简单明了.应用这种方法很容易证明克莱姆法则.对这种构造方法加以改进,给出了线性空间Rn中扩充一组正交基的新方法.     

Lyapunov exponents of free operators

Lyapunov exponents of a dynamical system are a useful tool to gauge the stability and complexity of the system. This paper offers a definition of Lyapunov exponents for a sequence of free linear operators. The definition is based on the concept of the extended Fuglede-Kadison determinant. We establish the existence of Lyapunov exponents, derive formulas for their calculation, and show that Lyapunov exponents of free variables are additive with respect to operator product. We illustrate these results using an example of free operators whose singular values are distributed by the Marchenko-Pastur law, and relate this example to C.M. Newman's “triangle” law for the distribution of Lyapunov exponents of large random matrices with independent Gaussian entries. As an interesting by-product of our results, we derive a relation between the extended Fuglede-Kadison determinant and Voiculescu's S-transform.     

Permanent Semigroups

A permanent semigroup is a semigroup of n × n matrices on which the permanent function is multiplicative. If the underlying ring is an infinite integral domain with characteristic p > n or characteristic 0 we prove that any permanent semigroup consists of matrices with at most one nonzero diagonal. The same result holds if the ring is a finite field with characteristic p > n and at least n²+n elements. We also consider the Kronecker product of permanent semigroups and show that the Kronecker product of permanent semigroups is a permanent semigroup if and only if the pennanental analogue of the formula for the determinant of a Kronecker product of two matrices holds. This latter result holds even when the matrix entries are from a commutative ring with unity.     

Computing the complete CS decomposition

An algorithm for computing the complete CS decomposition of a partitioned unitary matrix is developed. Although the existence of the CS decomposition (CSD) has been recognized since 1977, prior algorithms compute only a reduced version. This reduced version, which might be called a 2-by-1 CSD, is equivalent to two simultaneous singular value decompositions. The algorithm presented in this article computes the complete 2-by-2 CSD, which requires the simultaneous diagonalization of all four blocks of a unitary matrix partitioned into a 2-by-2 block structure. The algorithm appears to be the only fully specified algorithm available. The computation occurs in two phases. In the first phase, the unitary matrix is reduced to bidiagonal block form, as described by Sutton and Edelman. In the second phase, the blocks are simultaneously diagonalized using techniques from bidiagonal SVD algorithms of Golub, Kahan, Reinsch, and Demmel. The algorithm has a number of desirable numerical features.      

Positive definite completions of partial Hermitian matrices

The question of which partial Hermitian matrices (some entries specified, some free) may be completed to positive definite matrices is addressed. It is shown that if the diagonal entries are specified and principal minors, composed of specified entries, are positive, then, if the undirected graph of the specified entries is chordal, a positive definite completion necessarily exists. Furthermore, if this graph is not chordal, then examples exist without positive definite completions. In case a positive definite completion exists, there is a unique matrix, in the class of all positive definite completions, whose determinant is maximal, and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial Hermitian matrix. Additional observations regarding positive definite completions are made.     

Enumeration of symmetry classes of alternating sign matrices and characters of classical groups

An alternating sign matrix is a square matrix with entries 1, 0 and −1 such that the sum of the entries in each row and each column is equal to 1 and the nonzero entries alternate in sign along each row and each column. To some of the symmetry classes of alternating sign matrices and their variations, G. Kuperberg associate square ice models with appropriate boundary conditions, and give determinant and Pfaffian formulae for the partition functions. In this paper, we utilize several determinant and Pfaffian identities to evaluate Kuperberg's determinants and Pfaffians, and express the round partition functions in terms of irreducible characters of classical groups. In particular, we settle a conjecture on the number of vertically and horizontally symmetric alternating sign matrices (VHSASMs). Dedicated to the memory of David Robbins.     

A two-step method for estimating high-dimensional Gaussian graphical models

The problem of estimating high-dimensional Gaussian graphical models has gained much attention in recent years. Most existing methods can be considered as one-step approaches, being either regression-based or likelihood-based. In this paper, we propose a two-step method for estimating the high-dimensional Gaussian graphical model. Specifically, the first step serves as a screening step, in which many entries of the concentration matrix are identified as zeros and thus removed from further consideration. Then in the second step, we focus on the remaining entries of the concentration matrix and perform selection and estimation for nonzero entries of the concentration matrix. Since the dimension of the parameter space is effectively reduced by the screening step,the estimation accuracy of the estimated concentration matrix can be potentially improved. We show that the proposed method enjoys desirable asymptotic properties. Numerical comparisons of the proposed method with several existing methods indicate that the proposed method works well. We also apply the proposed method to a breast cancer microarray data set and obtain some biologically meaningful results.